Abstract

Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev polynomials equivalent to the formula

\frac{1}{1 − ax − x^{2}} ∗ \frac{1}{1 − bx − x^{2}} = \frac{1 − x^{2}}{1 − abx − (2 + a^{2} + b^{2})x^{2} − abx^{3} + x^{4}} ,

where ∗ denotes the Hadamard product. In a similar way, by considering tilings of a 2 × n rectangle with 1 × 1 and 1 × 2 bricks in the top row, and 1 × 1 and 1 × n bricks in the bottom row, we find an explicit formula for the Hadamard product

\frac{1}{1 − ax − x^{2}} ∗ \frac{x^{m}}{1 − bx − x^{n}} .